CFA Program Fundamentals eBook PDF

CFA® Program Fundamentals 3rd Edition Helping You Get Prepared For The CFA Program Curriculum CFA FUNDAMENTALS, 3RD EDITION ©2024 Kaplan, Inc. All rights reserved. Published in 2024 by Kaplan, Inc. These materials may not be copied without written permission from the author. The unauthorized du...

CFA® Program Fundamentals 3rd Edition Helping You Get Prepared For The CFA Program Curriculum CFA FUNDAMENTALS, 3RD EDITION ©2024 Kaplan, Inc. All rights reserved. Published in 2024 by Kaplan, Inc. These materials may not be copied without written permission from the author. The unauthorized duplication of these study tools is a violation of global copyright laws and the CFA Institute Code of Ethics. Your assistance in pursuing potential violators of this law is greatly appreciated. Required CFA Institute disclaimer: Kaplan Schweser is a CFA Institute Prep Provider. Only CFA Institute Prep Providers are permitted to make use of CFA Institute copyrighted materials which are the building blocks of the exam. We are also required to update our materials every year and this is validated by CFA Institute. CFA Institute does not endorse, promote, review or warrant the accuracy or quality of the product and services offered by Kaplan Schweser. CFA Institute®, CFA® and “Chartered Financial Analyst®” are trademarks owned by CFA Institute. Certain materials contained within this text are the copyrighted property of CFA Institute. Te following is the copyright disclosure for these materials: “Copyright, 2022, CFA Institute. Reproduced and republished from 2023 Learning Outcome Statements, Level I, II, and III questions from CFA® Program Materials, CFA Institute Standards of Professional Conduct, and CFA Institute’s Global Investment Performance Standards with permission from CFA Institute. All Rights Reserved.” Disclaimer: The Schweser Study Tools should be used in conjunction with the original readings as set forth by CFA Institute in their 2023 Level I CFA Study Guide. The information contained in these study tools covers topics contained in the readings referenced by CFA Institute and is believed to be accurate. However, their accuracy cannot be guaranteed nor is any warranty conveyed as to your ultimate exam success. The authors of the referenced readings have not endorsed or sponsored these Schweser study tools. CONTENTS Foreword......................................................................................... i Quantitative Methods..................................................................1 Economics................................................................................... 53 Financial Statement Analysis.................................................. 105 Corporate Issuers.................................................................... 189 Equity Investments.................................................................. 243 Fixed Income............................................................................ 283 Derivatives................................................................................ 319 Alternative Investments.......................................................... 345 Portfolio Management and Wealth Planning........................ 365 Ethical and Professional Standards....................................... 411 Index.......................................................................................... 417 This page has been left intentionally blank FOREWORD WHAT IS THIS EBOOK? This eBook was developed by CFA® Charterholders and exam prep experts for those interested in or currently enrolled in the CFA Program. The eBook contains foundational information about each curriculum topic and links to important dates and exam details. After each topic section, readers can apply their knowledge to practice questions and get detailed explanations to use during their CFA Program prep. CFA PROGRAM DETAILS The CFA Program is a series of three rigorous examinations created and administered by the CFA Institute. Candidates should expect to spend a minimum of four years in the program and study hundreds of hours for each exam. EXAM DATES The Level I exam is administered four times per year. Level II and Level III exams are administered three times a year. See Upcoming Dates EXAM DURATION Following are the CFA exam durations: Level I: Two equal 135-minute sessions Level II: Two equal 132-minute sessions Level III: Two equal 132-minute sessions ©2024 Kaplan, Inc. Page i  Foreword EXAM STRUCTURE For each CFA exam, candidates will sit for two sessions in one day: a morning session and an afternoon session. EXAM FORMAT The Level I and Level II exams are multiple-choice, formatted as questions, statements, charts, and tables. The Level II exam contains item set question types. The Level III exam is a combination of constructed response essays and multiple-choice questions. Learn About Exam Formats EXAM LOCATIONS Computer-based exams are offered in hundreds of CFA testing centers around the globe. After registering, candidates will be able to search by city and country for an available seat at an official CFA Institute testing center. EXAM RESULTS Level I and Level II exam results are sent to candidates approximately 5 to 8 weeks after the close of an exam window, depending on exam level. Level III candidates receive their results approximately 10 weeks after completing the exam. See Recent Pass Rates EXAM SCORING Candidates receive exam results that indicate either “pass” or “did not pass.” Candidates do not know how many points they scored in each section, but they do receive indications of whether they scored below 50%, between 50% and 70%, or above 70% in each section. Learn About Grading Page ii ©2024 Kaplan, Inc. Foreword START YOUR CURRICULUM PREP WITH MORE KAPLAN SCHWESER FREE RESOURCES For more than 30 years, Kaplan Schweser has been the premier provider of exam prep for the CFA exam. Start your free trial today, and get access to some of the best study tools we offer. Free Trial to Kaplan Schweser Study Materials Free CFA Program Question of the Day Free Monthly Candidate Guide Free Prep Product Samples READY TO REGISTER? Create an account with CFA Institute, and select which exam you’d like to take through the CFA Institute website. ©2024 Kaplan, Inc. Page iii  This page has been left intentionally blank Page iv ©2024 Kaplan, Inc.  Quantitative Methods TIME VALUE OF MONEY In this section, we will use timelines to calculate the present and future values of lump sums and annuities. Timelines will help you keep cash flows organized and allow you to see the timing of one cash flow in relation to the other cash flows and in relation to the present (i.e., today). Before we set up a timeline, however, let’s look at more of the terms we will utilize throughout the discussion. A lump sum is a single cash flow. Lump sum cash flows are one-time events and therefore are not recurring. An annuity is a finite number of equal cash flows occurring at fixed intervals of equal length over a defined period of time (e.g., monthly payments of $100 for three years). Present value is the value today of a cash flow to be received or paid in the future. On a timeline, present values occur before (to the left of) their relevant cash flows. Future value is the value in the future of a cash flow received or paid today. On a timeline, future values occur after (to the right of) their relevant cash flows. A perpetuity is a series of equal cash flows occurring at fixed intervals of equal length forever. The discount rate and compounding rate are the rates of interest used to find the present and future values, respectively. ©2024 Kaplan, Inc. Page 1  LUMP SUMS Future Value of a Lump Sum We’ll start our discussion with the future value of a lump sum. Assume you put $100 in an account paying 10% and leave it there for one year. How much will be in the account at the end of that year? The following timeline represents the one-year time period. Figure 1: Determining Future Value at t = 1 In one year, you’ll have $110. That $110 will consist of the original $100 plus $10 in interest. To set that up in an equation, we say the future value in one year, FV1, consists of the original $100 plus the interest, i, it earns. FV1 = $100 + (i × $100) Since $100 is the present value, the original deposit, we can substitute PV0 for the $100 in the equation. FV1 = PV0 + (i × PV0) Factoring PV0 out of both terms on the right side of the equation we are left with: FV1 = PV0(1 + i) (1) The result of these mathematical manipulations is the general equation for finding the future value of a lump sum invested for one year at a rate of interest i. Had it not been so easy to do the calculation in our heads, we would have substituted for the variables in the equation and gotten the following: FV1 = $100(1.10) = $110 What if you leave the money in the account for two years? After one year you’ll have $110, the original $100 plus $10 interest (or 10% times 100). At the end of the second year you will have the $110 plus interest on the $110 during the second year. The interest earned in the second year equals 10% times the $110 balance with which you began the year. Therefore, the interest earned in the second year consists of interest on the original Page 2 ©2024 Kaplan, Inc.  Quantitative Methods $100 plus interest earned in the second year on the interest earned during the first year but left in the account. When interest is earned or paid on interest, the process is referred to as compounding. This explains why future values are sometimes referred to as compound values. Now our timeline expands to include two years. Although the numbering is totally arbitrary and we could have used any number to indicate today, we are assuming we deposit $100 at time 0 on the timeline. We already know the value after one year, FV1, so let’s start there. Figure 2: Determining Future Value at t = 2 We know from Equation 1 that the future value of a lump sum invested for one year at interest rate i is the lump sum multiplied by (1 + i ). We simply find the future value of $110 invested for one year at 10% by using Equation 1 (adjusted for the different points in time), which gives us $121. FV1 = PV0(1 + i) FV2 = FV1(1 + i) FV2 = $110(1.10) = $121 To take this example a step further, we make some additional adjustments. We know from Equation 1 that FV1 is equal to PV0(1 + i ). Let’s further develop relationships between future and present value. We start with: FV2 = FV1(1 + i) Substituting, we get: FV2 = PV0(1 + i)(1 + i) And we end with: FV2 = PV0(1 + i)2 (2) Equation 2 is the general equation for finding the future value of a lump sum invested for two years at interest rate i. In fact, we have actually discovered the general relationship between the present value of a lump sum and its future value at the end of any number of periods, as long as the interest rate remains the same. We can state the general relationship as the following: FV2 = PV0(1 + i)2 (3) ©2024 Kaplan, Inc. Page 3  Quantitative Methods Equation 3 says the future value of a lump sum invested for n years at interest rate i is the lump sum multiplied by (1 + i)n. Let’s look at some examples. We’ll assume an initial investment today of $100 and an interest rate of 5%. Future value in 1 year: $100(1.05) = $105 Future value in 5 years: $100(1.05)5 = $100(1.2763) = $127.63 Future value in 15 years: $100(1.05)15 = $100(2.0789) = $207.89 Future value in 51 years: $100(1.05)51 = $100(12.0408) = $1.204.08 Regardless of the number of years, as long as the interest rate remains the same, the relationship in Equation 3 holds. Up to this point we have assumed interest was paid annually (i.e., annual compounding). However, most financial institutions pay and charge interest over much shorter periods. For instance, if an account pays interest every six months, we say interest is “compounded” semiannually. Every three months represents quarterly compounding, and every month is monthly compounding. Let’s look at an example with semiannual compounding. Again, let’s assume that we deposit $100 at time zero, and it remains in the account for one year. This time, however, we’ll assume the financial institution pays interest semiannually. We will also assume a stated or nominal rate of 10%, meaning it will pay 5% every six months. Figure 3: Future Values With Semi-Annual Compounding Page 4 ©2024 Kaplan, Inc.  Quantitative Methods In order to find the future value in one year, we must first find the future value in six months. This value, which includes the original deposit plus interest, will earn interest over the second six-month period. The value after the first six months is the original deposit plus 5% interest, or $105. The value after another six months (one year from deposit) is the $105 plus interest of $5.25 for a total of $110.25. The similarity to finding the FV in two years as we did in Equation 2 is not coincidental. Equation 2 is actually the format for finding the FV of a lump sum after any two periods at any interest rate, as long as there is no compounding within the periods. The periods could be days, weeks, months, quarters, or years. To find the value after one year when interest is paid every six months, we multiplied by 1.05 twice. Mathematically this is represented by: FV = $100(1.05)(1.05) = $100(1.05)2 FV = $100(1.1025) = $110.25 The process for semiannual compounding is mathematically identical to finding the future value in two years under annual compounding. In fact, we can modify Equation 2 to describe the relationship of present and future value for any number of years and compounding periods per year. m×n ⎛ i⎞ FVn = PV0 ⎜⎜1+ ⎟⎟⎟ (4) ⎜⎝ m⎠ where: FVn = future value after n years PV0 = present value i = stated annual rate of interest m = number of compounding periods per year m × n= total number of compounding periods (the number of years times the compounding periods per year) For semiannual compounding m = 2; for quarterly compounding m = 4; and for monthly compounding m = 12. If you leave money in an account paying semiannual interest for four years, the total compounding periods would be 4 × 2 = 8. Interest would be calculated and paid eight times during the four years. Let’s assume you left your $100 on deposit for four years, and the ©2024 Kaplan, Inc. Page 5  Quantitative Methods bank pays 10% interest compounded semiannually. We’ll use Equation 4 to find the amount in the account after four years. 4×2 ⎛ 0.10 ⎟⎞ FV = $100 ⎜⎜1+ ⎟ ⎜⎝ 2 ⎟⎠ FV = $100(1.05) 8 FV = $100(1.4775) = $1471.75 where: n = 4, because you will leave the money in the account for four years m = 2, because the bank pays interest semiannually i = 10% (the annual stated or nominal rate of interest) If the account only paid interest annually, the future value would be the following: 4 x1 ⎛ 0.10 ⎟⎞ FV = $100 ⎜⎜1+ ⎟ ⎜⎝ 1 ⎟⎠ FV = $100(1.10) 4 FV = $100(1.4641) = $146 1.41 The additional $1.34 (i.e., $147.75 – $146.41 = $1.34) is the extra interest earned from the compounding effect of interest on interest. Although the differences do not seem profound, the effects of compounding are magnified with larger values, greater number of compounding periods per year, or higher nominal interest rates. In our example, the extra $1.34 was earned on an initial deposit of $100. Had this been a $1 billion deposit, the extra interest differential from compounding semiannually rather than annually would have amounted to $13,400,000! To demonstrate the effect of increasing the number of compounding periods per year, let’s look at several alternative future value calculations when $100 is deposited for one year at a 10% nominal rate of interest. In each case, m is the number of compounding periods per year. m = 1 (annually) FV = $100(1.10) = $110 m = 2 (every 6 months) FV = $100(1.05)2 = $110.25 m = 4 (quarterly) FV = $100(1.025)4 = $110.38 m = 6 (every 2 months) FV = $100(1.0167)6 = $110.43 m = 12 (monthly) FV = $100(1.008333)12 = $110.47 m = 52 (weekly) FV = $100(1.001923)52 = $110.51 m = 365 (daily) FV = $100(1.000274)365 = $110.52 Page 6 ©2024 Kaplan, Inc.  Quantitative Methods You will notice two important characteristics of compounding: 1. For the same present value and interest rate, the future value increases as the number of compounding periods per year increases. 2. Each successive increase in future value is less than the preceding increase. (The future value increases at a decreasing rate.) Effective Interest Rates. The concept of compounding is associated with the related concept of effective interest rates. In our semiannual compounding example, we assumed that $100 was deposited for one year at 10% compounded semiannually. We represented it graphically using a timeline as follows: Figure 4: Effective lnterest Rates The stated (nominal) rate of interest is 10%. However, determining the actual rate we earned involves comparing the ending value with the beginning value using Equation 5. You can determine the actual or “effective” rate of return by taking into consideration the impact of compounding. Equation 5 measures the change in value as a percentage of the beginning value. V1 − V0 effective return = (5) V0 where: V0 = total value of the investment at the beginning of the year V1 = total value of the investment at the end of the year You will notice Equation 5 stresses using the values at the beginning and the end of the year (actually, any twelve month period). By convention, we always state effective interest rates in terms of one year. ©2024 Kaplan, Inc. Page 7  Quantitative Methods Returning to our previous example, let’s substitute our beginning-of-year and end-of-year values into Equation 5. Because the interest was compounded semiannually instead of annually, we actually earned 10.25% on the account rather than the 10% stated rate. V1 − V0 effective return = (5) V0 $110.25 − $100 effective return = = 0.1025 = 10.25% $100 Let’s employ algebraic principles and rewrite Equation 5 in the following form: V1 − V0 V V effective return = ⇒ 1− 0 V0 V0 V0 V1 effective return = −1 (6) V0 Now let’s restate Equation 6 in terms of FV and PV: m×1 V1 FV1 ⎛ i⎞ −1 is equivalent to −1 , and FV1 = PV0 ⎜⎜⎜1+ ⎟⎟⎟ V0 PV0 ⎝ m⎠ m×1 ⎛ i⎞ PV0 ⎜⎜1+ ⎟⎟⎟ FV1 ⎜⎝ m⎠ −1 can be rewritten as −1 (7) PV0 PV0 Remember, we always state effective returns in annual terms. Thus we can set n = 1 in the equation and the PV0 in the numerator and denominator cancel each other out. Substituting Equation 7 back into Equation 6 we get the following: m ⎛ i⎞ effective return = ⎜⎜1+ ⎟⎟⎟ −1 (8) ⎜⎝ m⎠ We have arrived at the general equation to determine any effective interest rate in terms of its stated or nominal rate and the number of compounding periods per year. Let’s investigate a few examples of calculating effective interest rates for the same stated interest at varying compounding Page 8 ©2024 Kaplan, Inc.  Quantitative Methods assumptions. Notice that the effective rate increases as the number of compounding periods increase. 1 ⎛ 0.12 ⎞⎟ m = 1 (annual compounding) ⎜⎜1+ ⎟ − 1 = (1.12)1 − 1 = 0.12 = 12% ⎜⎝ 1 ⎟⎠ 2 ⎛ 0.12 ⎞⎟ m = 2 (semiannual) ⎜⎜1+ ⎟ − 1 = (1.06)2 − 1 = 0.1236 = 12.36% ⎜⎝ 2 ⎟⎠ 4 ⎛ 0.12 ⎞⎟ m = 4 (quarterly) ⎜⎜1+ ⎟ − 1 = (1.03) 4 − 1 = 0.1255 = 12.55% ⎜⎝ 4 ⎟⎠ 12 ⎛ 0.12 ⎞⎟ m = 12 (monthly) ⎜⎜1+ ⎟ − 1 = (1.01)12 − 1 = 0.1268 = 12.68% ⎜⎝ 12 ⎟⎠ 365 ⎛ 0.12 ⎞⎟ m = 365 (daily) ⎜⎜1+ ⎟ − 1 = (1.0003288)365 − 1 = 0.1275 = 12.75% ⎜⎝ 365 ⎟⎠ Geometric Mean Return. The geometric mean return is a compound annual growth rate for an investment. For instance, assume you invested $100 at time 0 and that the investment value grew to $220 in 3 years. What annual return did you earn, on average? Using our future value formula from Equation 3: 100(1+i)3 = 220 (7) The interest rate in Equation 7 is the geometric mean or compound average annual growth rate earned on the investment. Solving Equation 7, gives i = 30%.1 More formally, the geometric mean is found using the following equation: FVn ⎛ ⎞1 n GM = n −1 or ⎜⎜ FVn ⎟⎟ −1 (8) PV ⎜⎝ PV ⎟⎠ where: GM = geometric mean FVn = future value of the lump sum investment PV = present value, or the initial lump sum investment n = number of years over which the investment is held 1 To solve this problem, divide both sides of the equation by 100, then take the third root and subtract 1; i = [220/100]0.33 –1. ©2024 Kaplan, Inc. Page 9  Quantitative Methods Thus, if you invested $500 in an mutual fund five years ago and now the original investment is worth $901.01, we would find the geometric mean return as follows: ⎛ $901.01⎞⎟1/5 ⎜ − 1 = (1.80202)1/5 − 1 = 1.125 − 1 = 12.5% ⎜⎝⎜ $500 ⎟⎠⎟ The mutual fund provided an average annual return of 12.5%. The geometric mean can also be stated using returns over several periods of equal length using the following formula: GM = n (1+ x1 )(1+ x 2 )(1+ x 3 )...(1+ x n ) – 1 (9) where: GM = geometric mean xi = ith return measurement (the first, second, third, etc.) n = number of data points (observations) We add 1 to each observation’s value, which is a percentage expressed as a decimal, multiply all the observations together, find the nth root2 of the product, and then subtract one. Let’s return to our mutual fund example. This time we will calculate the geometric mean return differently. Assume that over the last five years the fund has provided returns of 15%, 12%, 14%, 16%, and 6%. What was the geometric mean return for the fund? GM = n (1+ x1 )(1+ x 2 )(1+ x 3 )...(1+ x n ) −1 GM = 5 (1 + 0.15) × (1 + 0.12) × (1 + 0.14 ) × (1 + 0.16) × (1 + 0.06) −1 = 5 (1.15)(1.12)(1.14)(1.16)(1.06) − 1 = 5 1.80544 − 1 = 0.125 GM = 12.5%3 The geometric mean shows the average annual growth in your cumulative investment for the five years, assuming no funds are withdrawn. In other words, the geometric mean assumes compounding. In fact, when evaluating investment returns, the geometric mean is often referred to as the compound mean. 2 nth refers to the number of observations. If there were two observations, you would find the square root. With twenty observations you find the 20th root. 3 You will notice that, although they are close, the geometric mean is smaller than the arithmetic mean. In fact, the geometric will always be less than or equal to the arithmetic mean. Page 10 ©2024 Kaplan, Inc.  Quantitative Methods Present Value of a Lump Sum Recall that Equation 3 showed us the relationship between the present and future values for a lump sum. FVn = PV(1 + i)n (3) In Equation 3 the future value is determined by multiplying the present value by (1 + i)n. To solve for the present value, we can divide both sides of the equation by (1 + i)n. FVn PV = (3) (1+ i) n Finding a present value is actually deducting interest from the future value, which we refer to as discounting. The present value can be viewed as the amount that must be invested today in order to accumulate a desired amount in the future. The “desired amount in the future” is known as the “future value.” Returning to the future value examples used earlier, we can demonstrate how to calculate present values. We will assume the same discount rate of 5%. (FV) The value in 1 year of $100 deposited today: $100(1.05) = $105 (PV) The value today of $105 to be received in 1 year: $105 $100 = (1.05) (FV) The value in 5 years of $100 deposited today: $100(1.05)5 = $127.63 (PV) The value today of $127.63 to be received in 5 years: $127.63 $100 = (1.05)5 (FV) The value in 15 years of $100 deposited today: $100(1.05)15 = $207.89 (PV) The value today of $207.89 to be received in 15 years: $207.89 $100 = (1.05)15 ©2024 Kaplan, Inc. Page 11  Quantitative Methods (FV) The value in 51 years of $100 deposited today: $100(1.05)51 = $1,204.08 (PV) The value today of $1,204.08 to be received in 51 years: $1, 204.08 $100 = (1.05)51 ANNUITIES Recall that an annuity is a series of equal payments that occur at fixed intervals of equal length through time. The series may consist of two or more cash flows. Future value of an annuity due. Assume the cash flows represent deposits to an account paying 10%, and you want to know how much you will have in the account at the end of the fifth year. Point zero on the timeline is when the first deposit is made. Each successive deposit is made at the beginning of each year, so the last deposit is made at the beginning of year five. When cash flows come at the beginning of the period, the annuity is known as an annuity due. An annuity due is typically associated with leases or other situations where payments are made in advance of services or products received or rendered. Our goal is to determine the amount we will have in the account at point five (i.e., the end of year five). Consider an equipment lease. The user of the equipment (the lessee) pays the owner of the equipment (the lessor) a fee for the right to use the equipment over the next period. That is, leases are prepaid. An example of an ordinary (or deferred) annuity is a mortgage loan (or any debt) with end of period payments. When the money is borrowed, the borrower makes payments at the end of set periods, usually every month or semi-annually. Each payment will include interest, which reflects the cost of the money over the previous period. When a mortgage is fully amortized, each of the fixed payments includes interest on the outstanding principal as well as a partial repayment of principal. Since the outstanding principal decreases with each payment, the proportion of each successive payment representing interest decreases, and the proportion representing principal increases. The process of finding the future value of an annuity in this manner is equivalent to summing the future value of each individual cash flow. Each cash flow is assumed to earn a 10% return for each of the indicated number of years. For example, the $100 at time 0 will remain on deposit for a total of five years, so we multiply $100 by (1 + i)5 and obtain $161.05. When all the cash flows are compounded to find their future values, we add them to find the total future value of the five cash flows, which totals $671.56. In Page 12 ©2024 Kaplan, Inc.  Quantitative Methods other words, if you deposit $100 per year in an account paying 10%, you will have $671.56 in five years. Figure 5: Future Value of an Annuity Due Future value of an ordinary annuity. Now we’ll consider the same annuity of five $100 deposits, but we’ll assume that the deposits occur at the end of each year. When cash flows are at the end of the period, the annuity is known as an ordinary annuity. An ordinary annuity represents the typical cash flow pattern for loans, such as those for automobiles, homes, furniture, fixtures, and even businesses. You will notice that the future value of the ordinary annuity is less than the future value of the annuity due, although the number and amount of the deposits is the same. The difference results from the fact that the deposits for the annuity due are all received exactly one period earlier than the corresponding deposits for the ordinary annuity. Each deposit for the annuity due accrues an additional period’s (in this case, a year’s) interest. The value of an annuity due is equal to the value of an ordinary annuity plus one period’s interest. In our example the length of the period is a year, so $610.51 + ($610.51 × 0.10) = $671.56. Looking at the relationship form another perspective, you will notice that the four deposits at points one though four are exactly the same for both annuities. The difference between the two types of annuities is the treatment of the remaining deposit. With the annuity due (Figure 5), the remaining deposit is made at time zero and earns interest for five years. ©2024 Kaplan, Inc. Page 13  Quantitative Methods With the ordinary annuity (Figure 6), the remaining deposit is made at the end of year five immediately before the account is closed and the money is withdrawn. Since the last deposit in the case of the ordinary annuity earns no interest, the difference between the future values of the two annuities must be the interest earned on the deposit made at time 0 in the case of the annuity due. The future value of the ordinary annuity is $610.51. The future value of the annuity due is $671.56, exactly $61.05 larger. Figure 6: Ordinary Annuity The solutions presented for both annuities were more for demonstration purposes than for actually calculating the future values. If you find yourself needing to calculate the future value of an annuity, you will use one of two other approaches: a financial calculator or a formula. Let’s illustrate the use of a financial calculator to find the future value of an annuity. Our example is based upon the keystrokes required for a Texas Instruments (TI) Business Analyst II Plus calculator. Ordinary annuity: Your calculator should be set to end of period payments and one payment per year. To set to end of period payments, press 2nd ⟹ BGN and press 2nd ⟹ SET until END is displayed, then 2nd ⟹ QUIT. (Since end is default, the display will not indicate end of period payments.) Page 14 ©2024 Kaplan, Inc.  Quantitative Methods To set to one payment per year, press 2nd ⟹ P/Y ⟹ 1 ⟹ ENTER, 2nd ⟹ QUIT. The keystrokes4 to find the future value are the following: –100 PMT [The calculator assumes one of the payments is an outflow (from the perspective of the investor) and one is an inflow. The negative sign indicates an outflow (the deposit)]. 5 N 10 I/Y CPT FV = $610.51 Annuity due. For an annuity due, set the financial calculator for beginning- of-period payments and one payment per year. To set to one payment per year, press 2nd ⟹ P/Y ⟹ 1 ⟹ ENTER, 2nd ⟹ QUIT. To set to end-of- period payments, press 2nd ⟹ BGN and press 2nd ⟹ SET until BGN is displayed, then 2nd ⟹ QUIT. (BGN will show in the calculator display.) The keystrokes to find the future value are the following: –100 PMT [The calculator assumes one of the payments is an outflow (from the perspective of the investor) and one is an inflow. The negative sign indicates an outflow (the deposit)]. 5 N 10 I/Y CPT FV = $671.56 While using a financial calculator is the recommended procedure, you can also use a formula for calculating the future value of an annuity. To find the future value of an ordinary annuity, multiply the cash flow (payment) by the formula and the result is the future value. Assume the same $100 deposits made each year for five years earning a 10% return, compounded annually. (1+ i) n −1 FVn = PMT i (1.10)5 −1 FV5 = $100 0.10 ⎛1.6105 −1⎞⎟ FV5 = $100 ⎜⎜ = $100(6.1051) = $610.51 ⎜⎝ 0.10 ⎟⎟⎠ 4 Calculator Tip: Note that PV, FV, PMT, I/Y, and N are just memory registers. If there are values in these registers from previous computations, you will need to clear them. There are two ways to do this. The brute force method is to simply put a zero into the register that you’re not using. The second and more elegant method is to press 2nd ⟹ CLR TVM to completely clear the time value of money register before you enter the next set of data points. ©2024 Kaplan, Inc. Page 15  Quantitative Methods The same process would be used to calculate the future value of an annuity due. We multiply the future value of the ordinary annuity by (1 + i) to adjust for the fact that each cash flow is received one period earlier, and thus earns an extra period’s interest, with the annuity due. ⎛1.6105 −1⎟⎞ FV5 = $100 ⎜⎜ = $100(6.1051) = $610.51(1 + 0.10)) ⎜⎝ 0.10 ⎟⎟⎠ = $671.56 Present value of an annuity due. When we found the future value of an annuity, we compounded each cash flow individually and summed them at a future date. To find the present value of an annuity, we discount all the future values and sum them up. We’ll start with the annuity due we used before. The cash flows are assumed to be paid/received at the beginning of each year, and we want to find the aggregate present value of the five cash flows at point zero on the timeline. Again we assume an interest rate of 10%. The process of finding the present value of an annuity due is equivalent to summing the present value of each individual cash flow. The cash flow stream is illustrated in Figure 7. Each cash flow is discounted at a 10% rate for each of the indicated number of years. For example, the present value of $100 at time 4 equals 100 divided by (1 + 0.10)4 = $68.30; the present value of $100 at time 3 equals 100 divided by (1 + 0.10)3 = $75.13, and so on. When all the cash flows are discounted to find their present values, we add them to find the total present value of the five cash flows, which totals $416.99. Figure 7 illustrates the calculation of the present value of an annuity due. Figure 7: Present Value of an Annuity Due Page 16 ©2024 Kaplan, Inc.  Quantitative Methods To find the present value of this annuity due using a TI BA II Plus calculator, you would make the following key entries (be sure to turn on BGN mode): –100 PMT 5 N 10 I/Y CPT PV = $416.99 There are several ways of interpreting the $416.99. The $416.99 is the present value of the five $100 payments/receipts, but what does “present value” really mean? A simple interpretation is that if you put $416.99 in an account paying 10% interest, you will be able to withdraw $100 per year for five years. Another somewhat more sophisticated interpretation is that $416.99 is the maximum you would pay for an investment paying $100 per year with a required return of 10%. A third interpretation is that if you borrow $416.99 to be paid in five equal annual payments, you will pay $100.00 per payment. Regardless, assuming a 10% interest rate, $416.99 today is equivalent to five annual $100 cash flows, the first cash flow occurring today. Present value of an ordinary annuity. Likewise, the present value of an ordinary annuity may be calculated by summing the present value of each individual payment. Figure 8 illustrates the cash flow stream for the ordinary annuity. In this case the first cash flow occurs one year from today with the other four coming yearly after that. The present value of each cash flow is found and then added to the others to get a total of $379.08. ©2024 Kaplan, Inc. Page 17  Quantitative Methods Figure 8: Present Value of an Ordinary Annuity The keystrokes to find the present value of an ordinary annuity are [be sure to turn BGN mode off (i.e., you are in END mode)]: –100 PMT 5 N 10 I/Y CPT PV = $379.08 In addition, we may calculate the present value of an ordinary annuity using the following formula: ⎡1 1 ⎤ PVA = PMT ⎢⎢ − n⎥ ⎥ ⎣ i i(1+ i) ⎦ ⎡ 1 1 ⎤ PVA = $100 ⎢⎢ − ⎥ 0. 10 ( ) 5⎥ ⎣ 0. 10 1. 10 ⎦ PVA = $100[10 − 6.2092] PVA = $100[3.7908] = $379.08 Page 18 ©2024 Kaplan, Inc.  Quantitative Methods Thus we are able to confirm the present value of the ordinary annuity is $379.08.5 When both principal and interest are included in a loan payment (rather than a series of interest payments followed by the repayment of principal at the maturity of the loan), we say the loan is fully amortized. Let’s use the ordinary annuity of $379.08 as an example. Let’s assume you have borrowed $379.08 for the purchase of a household item and have agreed to pay for it in five equal payments at 10% interest. We know from the previous example that the payments will be $100 each. Figure 9 shows how each payment would be broken down into interest and principal if the loan were fully amortized. Figure 9: Amortized Loan (amortization of $379.08 for five years at 10%) Payment Interest Principal Balance $379.08 1 $100 37.91 62.09 316.99 2 $100 31.70 68.30 248.69 3 $100 24.87 75.13 173.56 4 $100 17.36 82.64 90.92 5 $100 9.09 90.91 0* *slight rounding error Each of the five payments repays a portion of the principal borrowed and pays interest on the balance remaining after the previous payment. The first payment includes 10% interest on the entire loan amount of $379.08, or $37.91. The remainder of the payment ($100 – $37.91 = $62.09) is applied to the principal, leaving a balance of $316.99. The second payment includes 10% interest on the new balance of $316.99, or $31.70. Again, the remainder of the payment, $68.30, is applied to the principal. This process continues until the loan is fully paid. You can see in Figure 9 that the interest in each payment decreases while the principal increases. Present value of a perpetuity. As explained earlier, a perpetuity is a series of equal cash flows occurring at the same interval forever. The present value of a perpetuity equals the periodic cash amount divided by the discount (interest) rate. For example, assume you own a stock paying $2 dividends forever. Further, assume the appropriate discount rate on the stock is 10%. The present value of the perpetuity equals $2 divided by 10%, which equals $20. In other words, you should be willing to pay $20 for a stock 5 Similar to calculations related to future value, the present value of an annuity due can be calculated from the present value of the ordinary annuity by multiplying the present value of the ordinary annuity by (1 + i). ©2024 Kaplan, Inc. Page 19  Quantitative Methods expected to pay $2 dividends per year forever, using a discount rate of 10%. More formally, we can state the formula for finding the present value of a perpetuity as follows: CF PVperpetuity = i where: CF = periodic cash flows of the perpetuity i = discount rate INTRODUCTION TO STATISTICS Webster’s Collegiate Dictionary6 defines statistics as “a branch of mathematics dealing with the collection, analysis, interpretation, and presentation of numerical data.” When we are trying to determine certain characteristics about a large population, we take a sample from that population. We then use that sample to derive certain statistics and make inferences about the population. Population A population is the collection of all possible individuals, objects, measurements, or other items; (e.g., the population of the U.S. is all people who call the U.S. their home country). The population of carp in the world is not just those that live in beautifully maintained and landscaped pools in Japan. The carp population would include every single carp, no matter where it lives. An important characteristic of a population is its size. For instance, there are approximately 300 million people living in the U.S. That means there are about 300 million members of that particular population. If you wanted to estimate the average height of an American, you would not want to go around to every single person and measure his or her height. Even if you could afford the extreme cost in both time and money, it would be a logistical nightmare. How about weighing every carp in the world to find the average weight of an adult carp? The field of statistics allows us to estimate these values without actually measuring each member of the population. 6 Merriam Webster’s Collegiate Dictionary, 10th Edition, 2000. Page 20 ©2024 Kaplan, Inc.  Quantitative Methods Sample A sample is a portion or subset of a population that is used to estimate characteristics of (i.e., make inferences about) the population. If we were interested in the average height of a U.S. citizen, we could select people from all over the U.S. (a sample of people), measure them, and find the average height. The average height of the individuals in the sample is then used to infer the average height of all people in the United States. An important characteristic of samples is randomness. You don’t want to force the sample to yield statistics that are biased because of the way the sample is taken. For example, if you are trying to estimate the percentage of people in the U.S. who are over age 65, you would not take your sample observations7 from a retirement community. If you were to do that, you could estimate that nearly 100% of the U.S. population is over 65! An appropriate sample would be drawn from many different areas across the country in a random, unbiased way. Another characteristic important to the sample is size. When an extremely large sample is drawn, the costs can be very high, and the inferences not significantly stronger than those of a somewhat smaller sample. However, if the sample is too small, the inferences drawn from the sample may not be trustworthy. Even though we will not pursue ideal sample size in this chapter, it is important to remember that sample size is important to the value (the confidence) you can place on the inferences you make about a population. Variables A variable is an unknown quantity (measurement) that can have different values. For example, if you were estimating average height and measured every person in your sample, the first value of the variable “height” would be the height of the first person measured. The second value would be the height of the second person; the third value would be the height of the third person, and so on. The variable “height” would have as many values (observations) as there are people in your sample. For example, we might use the letter x to denote height. x1 is the notation used to denote the height of the first person sampled. x2 is the notation used to denote the height of the second person sampled, etc. For instance, if the first person is 70 inches tall, then x1 = 70. If the second person is 71 inches tall, then x2 = 71. 7 An observation is one member of a sample. A sample of size 30 has 30 observations. ©2024 Kaplan, Inc. Page 21  Quantitative Methods There are two main categories of variables: qualitative and quantitative. A qualitative variable measures attributes. These could include gender, religious preference, eye color, type of running shoe preferred, and place of birth. In other words, qualitative variables do not use numbers. Conversely, quantitative variables are expressed numerically. These could include the average number of children in the typical household, the average height of American females, the percentage of people in the population with false teeth, or the average number of computers sold daily. Quantitative variables can be categorized as discrete or continuous. Think of discrete as meaning that the variable can only have a countable number of easily identified values. If the variable can only take on a whole number value from 1 to 10, it would be considered discrete. Its only possible values are 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. You’ll notice that you can easily count each possible value. Now let’s say the variable can have any incremental value between 1 and 10. In this case the variable can assume an infinite number of possible values and is called a continuous variable. When the variable was discrete, two of its possible values were three and four, and it could not have a value between three and four. As a continuous variable, it can have the values three and four, but it can also take on any of the infinite values between three and four. To illustrate, consider a continuous variable that is measured in inches. If the value of the first measurement is 3.0 inches, the next possible value could be so close to 3.0 that we have no instrument capable of measuring the distance. Consider the maximum number of zeros you can place between the decimal point and a one. An example would be 3.00001, but of course you can place many more than four zeros between the decimal and the one. In fact, the number of zeros you could place between the decimal point and the one is infinite. An example of a discrete variable is the outcome of the roll of a die (1, 2, 3, 4, 5, or 6). An example of a continuous variable is the amount of rainfall during July in a city. Let’s turn our attention to the terminology that is used to describe data. If we want to estimate the average height of a large group of people, we Page 22 ©2024 Kaplan, Inc.  Quantitative Methods can measure a sample of them and write down each measurement.8 Let’s assume we collect the following measurements: Person Height in Inches 1 70 2 71 3 73 4 66 5 62 6 70 These measurements are referred to collectively as the sample data (plural), while each individual measurement or observation is a data point. One observation would be the value 70 inches. Another would be 66 inches. Once we have collected our data, we will look for ways of describing them as a whole (i.e., we will describe the distribution of the data). Frequency Distribution Often, we do not want to see all the data points (especially if the sample size is large) but rather are interested in seeing merely a summary of the data. One popular way to summarize the data is to tabulate the frequency of observations falling in various categories. The tally of observations falling in equally spaced intervals is called the frequency distribution of the data. The frequency distribution shows how the data are scattered. For instance, consider the following frequency distribution. Height Interval Frequency Tally (inches) 64 up to 66 10 66 up to 68 35 68 up to 70 40 70 up to 72 10 72 up to 74 5 The table indicates that from our sample of 100 individuals, 10 individuals are 64 up to 66 inches tall; 35 individuals are 66 up to 68 inches tall, 8 We are assuming a small sample size to make the example easier. In reality, you would measure anywhere from several dozen to several thousand people, depending on the size of the population. ©2024 Kaplan, Inc. Page 23  Quantitative Methods et cetera. One way to summarize the data visually is by graphing the frequency distribution as follows: Figure 10: Frequency Distribution Histogram 40 35 Frequency 10 10 5 64 to 66 66 to 68 68 to 70 70 to 72 72 to 74 Height A graph of a frequency distribution is called a histogram. Notice how the histogram illustrates how the data are scattered. There is a center (at the 3rd category) with a pattern around the center. In this example, the distribution is skewed, meaning that there are not an equal number of observations on either side of the middle interval. Alternatively, a symmetric distribution is one in which there are an equal number of observations on either side of the middle interval (i.e., follows a bell-shape). To describe the data further, we often want to find the center point of the data. The center point of the data is known as the central tendency of the distribution. There are a few statistical measures of central tendency, the most popular of which are the mean, median, and mode. MEASURES OF CENTRAL TENDENCY Mean Mean is just another word for average, or the center of the data. For instance, when we previously mentioned estimating the average height, we could have used the expression “mean height.” The most common measure of central tendency is the arithmetic mean. Page 24 ©2024 Kaplan, Inc.  Quantitative Methods The arithmetic mean for a sample is denoted by:9 x1 + x 2 +... + x n x= n Earlier, to estimate the average height of the U.S. population, we took a sample of people and measured their heights. By substituting the data (individual measurements) into the equation to find the sample mean, we get the following: 70 + 71+ 73 + 66 + 62 + 70 x= = 68.7 inches 6 Median The median is the middle observation of the ranked data. If we ranked our height observations from the smallest to the largest, the same number of observations will fall above and below the median value. Our observations are 62, 66, 70, 70, 71, and 73. The median of the six observations falls between the third- and fourth-ranked observations. Thus, the simple average of the third and fourth observations, 70, represents the median of our sample. There are two observations greater than 70 (71 and 73), and there are two observations less than 70 (62 and 66). A note is necessary at this point. You probably questioned why the mean and median, both measures of the center of the distribution, are different. In fact, you might even be asking yourself, “Why do we need so many measures of the center of the distribution in the first place?” This question deserves some attention. The median finds the center of the distribution by number of observations. There are an equal number of observations above and below the median, regardless of their values. The mean, on the other hand, actually adds all the observations together and divides by the number of observations to find the mathematical center. You probably noticed that the mean of the sample observations, 68.7, isn’t even one of the observations. This is quite frequently the case. The reason we calculate both the mean and median of a data set is to get an idea of where the true center lies. Since the median and mean find the center in different ways, a more accurate estimation of where the center is and what is influencing its location can be gained by observing both the mean and median. 9 Sometimes the summation symbol, Σ, is used to define the arithmetic mean as Σx/n. Also, the Greek symbol μ is often used to represent the mean of the entire population; x is used when deriving the mean of a sample. The method of calculating the population and sample means is the same, however. ©2024 Kaplan, Inc. Page 25  Quantitative Methods Mode The mode is the observation that appears most often. In our case both the median and the mode are 70. The mean, median, and mode are all measures of central tendency. They all locate the center of the observations or population. MEASURES OF DISPERSION Measures of central tendency show the location of the center of the distribution. We often also want a measure of how spread out or dispersed the data are. There are several popular measures of dispersion. Here, we will discuss the range, mean absolute deviation, and standard deviation. Range The range is simply the “distance” between the lowest and the highest observed values. In our case we say the observations ranged from 62 inches to 73 inches, or 11 inches. The larger the range, the wider the dispersion of observations, and the smaller the range, the narrower the dispersion of the data. We now have information about the central tendency and the dispersion of the data. We can look at these statistics together and learn much about the sample. For instance, we see its center is around 70 inches, and there is a range of 11 inches between the smallest and largest observations in the sample. A weakness of the range is that it uses only two data points. In contrast, the mean absolute deviation, variance, and standard deviation use all the data points. Mean Absolute Deviation The mean absolute deviation is a measure of the dispersion of the sample observations around the center of the distribution. It measures the average deviation from the mathematical mean. A deviation is measured as the distance from the mean to each observation. Page 26 ©2024 Kaplan, Inc.  Quantitative Methods With a mean of 68.7”, the deviations for our sample are: 70.0” – 68.7” = 1.3” 71.0” – 68.7” = 2.3” 73.0” – 68.7” = 4.3” 66.0” – 68.7” = –2.7” 62.0” – 68.7” = –6.7” 70.0” – 68.7” = 1.3” and the mean absolute deviation, MAD, can be calculated using the following: MAD = ∑ x−x n where: x = value of each observation x = arithmetic mean of the observations n = number of observations x – x = absolute value of each deviation10 70 − 68.7 + 71− 68.7 + 73 − 68.7 + 66 − 68.7 + 62 − 68.7 + 70 − 68.7 MAD= 6 and 1.3 + 2.3 + 4.3 + 2.7 + 6.7 +1.3 MAD = = 3.1 6 The mean absolute deviation for our sample is 3.1 inches. Therefore, on average, the sample observations fall 3.1 inches from the sample mean. If the mean absolute deviation had been 1.0 inch, the observations would be much more closely grouped around the mean, or less dispersed. Had the mean absolute deviation been 6 inches, the observations would be more spread out or dispersed. Variance and Standard Deviation Another way to measure the dispersion of our sample is with the variance and standard deviation. Both the variance and standard deviation are measures of dispersion of the data around the mean of a distribution. The calculation of the variance is similar to the mean absolute deviation 10 Since summing the negative and positive deviations would cancel them out, we have to ignore their signs and sum their absolute values. ©2024 Kaplan, Inc. Page 27  Quantitative Methods calculation. Once again we begin with the deviations of each data point from the mean. Instead of averaging the deviations, however, we average the squared deviations. This also gives us all positive numbers because squaring a negative number results in a positive product. Look at the following equation, which gives us the population variance: 2 σ2 = ∑ (X − μ ) N where: σ2 = population variance (the arithmetic mean of the squared deviations from the population mean) μ = mean of the population X = individual member of the population N = number of observations in the population After we calculate the variance for the population, we can find the standard deviation. The standard deviation, σ, is simply the square root of the variance, σ2. σ = σ2 or 2 σ= ∑ (X − μ ) N If we assume that our earlier data set of heights for six people included the entire population, the standard deviation would be calculated as follows: (70 − 68.7 ) 2 + (71− 68.7 ) 2 + (73 − 68.7 ) 2 + (66 − 68.7 ) 2 + (62 − 68.7) 2 + (70 − 68.7 ) 2 6 1.69 + 5.29 +18.49 + 7.29 + 44.89 +1.69 σ= = 13.22 = 3.64 6 Thus, the standard deviation of our population is 3.64 inches. The equations for variance and standard deviation of a sample are slightly different than those for the population. Since a sample is smaller than the population, finding the arithmetic mean of the squared deviations tends Page 28 ©2024 Kaplan, Inc.  Quantitative Methods to underestimate the true value. We have to adjust the formula slightly to account for this effect. The variance for a sample is as follows: s2 = ∑ (x − x)2 n −1 where: s2 = sample variance x = individual observation in the sample x = sample mean (n – 1)= number of sample observations minus 1 Notice we divided by (n – 1) rather than n. This is the modification for the size of the sample compared to the population. As with the population, the standard deviation of the sample is the square root of its variance. s = s2 or s2 = ∑ (x − x)2 n −1 For our previous example related to estimating the average height of a large population of people, we calculate the standard deviation as follows: (70 − 68.7 ) 2 + (71− 68.7 ) 2 + (73 − 68.7 ) 2 + (66 − 68.7 ) 2 + (62 − 68.7) 2 + (70 − 68.7 ) 2 6 −1 1.69 + 5.29 +18.49 + 7.29 + 44.89 +1.69 s= = 3.98 5 Thus, the standard deviation of our sample is 3.98 inches. The larger the standard deviation, the greater the dispersion of the observations about the mean. This means the observations are relatively spread out. This is often referred to as a “loose” distribution. The smaller the standard deviation, the more the observations are grouped around the mean. The observations are less spread out. We would describe this as a “tight” distribution. ©2024 Kaplan, Inc. Page 29  Quantitative Methods THE NORMAL DISTRIBUTION AND THE EMPIRICAL RULE FOR PROBABILITIES Used extensively in financial analysis, the normal distribution is the famous “bell-shaped curve” of which statisticians and others speak. In a normal distribution, the mean, median, and mode are equal. Although the normal distribution may not precisely describe the pattern of returns for common stocks, for example, it is commonly used because of its statistical properties, especially symmetry. Perfect symmetry implies: ◼ The center point of the distribution is the mean, median, and mode. ◼ 50% of all possible returns are to the right of (above) the mean ◼ 50% of all possible returns are to the left of (below) the mean. Figure 11 shows the normal distribution. Of particular interest are the symmetry of the distribution and the counting of standard deviations from the mean. To the right of (above) the mean, lines are drawn that would approximate the returns within one and two standard deviations of the mean. Symmetry implies we can include lines on the other side of (left of) the mean that are the same distance from the mean as their counterparts to the right. Figure 11: Normal Distribution In many business settings, we wish to estimate the likelihood of certain events (e.g., likelihood of positive earnings, of debt rating change, etc). A helpful rule for estimating the likelihood of events using the normal distribution is that approximately 68% of the data lie within 1 standard Page 30 ©2024 Kaplan, Inc.  Quantitative Methods deviation of the mean, and 95% of the data lie within 2 standard deviations of the mean. For instance, assume you have been hired to manage a portfolio for a client. Assume your portfolio has an expected mean annual return of 10% and a standard deviation of 5%. Your client asks you to estimate the probability that she will lose money next year on your portfolio. You can use the empirical rule to answer the question. For instance, the empirical rule states that there is approximately a 95% chance that next year’s portfolio return will lie within 2 standard deviations of the mean. Since the mean is 10% and the standard deviation is 5%, then 95% of the time the annual return will range from 0% to 20% (i.e., the mean return plus and minus 2 standard deviations). The remaining 5% of the time, the portfolio return will lie either below 0% or above 20%. If we assume the portfolio returns are symmetric (equal probabilities for returns above and below the mean), there is a 2.5% chance the return will lie below 0% and another 2.5% chance the return will lie above 20%. This client should feel pretty safe about her investment: There is only a 2.5% chance she will lose money next year. ©2024 Kaplan, Inc. Page 31  Quantitative Methods SUMMARY TIME VALUE OF MONEY A. A lump sum is a single cash flow. 1. The present value of a lump sum is the equivalent value today of a lump sum to be received or paid sometime in the future. a. As interest rates increase (decrease), present values decrease (increase). b. As the number of periods increase (decrease), present values decrease (increase). 2. The future value of a lump sum is the equivalent value in the future of a lump sum to be received or paid today. a. As interest rates decrease (increase), future values decrease (increase). b. As the number of periods increase (decrease), future values increase (decrease). B. The effective interest rate (i.e., return) depends upon the nominal rate and the number of compounding periods per year. As the number of compounding periods increases, the effective rate increases. C. The geometric mean return is a compound annual growth rate for an investment. Geometric mean return takes into account the effects of compounding. D. An annuity is a countable number of equal cash flows occurring at fixed intervals of equal length over a defined period of time. 1. The future value of an annuity is the equivalent lump sum value in the future of a series of cash flows to be received or paid. As interest rates decrease (increase), the future value of an annuity decreases (increases). 2. The present value of an annuity is the equivalent lump sum value today of a series of cash flows to be received or paid. As interest rates increase (decrease), the present value of an annuity decreases (increases). E. When loan payments include both principal and interest, we say the loan is amortized. With each successive payment, the proportion of the payment applied to interest decreases, and the proportion applied to principal increases. F. The present value of a perpetuity (an annuity that never ends) is the cash flow divided by the discount rate. Page 32 ©2024 Kaplan, Inc.  Quantitative Methods INTRODUCTION TO STATISTICS A. A population is the collection of all possible individuals, objects, or other items (e.g., all the people in a country). B. A sample is an unbiased, randomly selected representative portion of the population that is used to estimate characteristics of the population. C. Qualitative variables measure attributes (e.g., eye color, gender). D. Quantitative variables are expressed numerically, (e.g., the number of children per household). E. Quantitative variables can be discrete or continuous. 1. Discrete quantitative variables are countable. 2. Continuous quantitative variables have an infinite number of possible values. F. The arithmetic mean is the simple average of the values (i.e., the sum of the values divided by the number of values). x + x 2 +... + x n x= 1 n G. The median is the middle observation of the ranked data. H. The mode is the observation that occurs the most. I. The range is the distance between the largest and smallest data points. J. The mean deviation is the average distance from the mean. K. The variance and standard deviation are measures of dispersion of the data around the mean of the distribution. 1. Standard deviation equals the square root of the variance. 2. The population variance equals the average squared deviation from the mean: 2 σ2 = ∑ (X − μ ) N 3. The sample variance is similar to the population variance but has n – 1 in the denominator instead of N: s2 = ∑ (x − x)2 n −1 L. The empirical rule states that 68% of the data will lie within 1 standard deviation of the mean and that 95% of the data will lie within 2 standard deviations of the mean. This rule can be used to construct confidence intervals. WAS THIS INFORMATION HELPFUL? Yes No ©2024 Kaplan, Inc. Page 33  Quantitative Methods PRACTICE QUESTIONS: QUANTITATIVE METHODS Note: All dollar values are rounded to the nearest whole dollar. 1. For a given present value and interest rate, the future value: A. increases as the number of compounding periods per year increases. B. decreases as the number of compounding periods per year increases. C. remains the same as the number of compounding periods per year increases. D. remains the same as the number of compounding periods per year decreases. 2. For a given future value and interest rate, the present value: A. increases as the number of compounding periods per year increases. B. decreases as the number of compounding periods per year increases. C. remains the same as the number of compounding periods per year increases. D. remains the same as the number of compounding periods per year decreases. 3. Jim Wilson is planning to purchase a high performance sports car for $100,000. He will finance the purchase with a 5-year fully amortized loan at an interest rate of 5.0% with payments due at the end of each year. What is the interest portion of the payment in year three and the remaining principal balance at the end of year three? Interest Principal A. $5,000 $42,948 B. $3,145 $30,708 C. $5,000 $30,708 D. $3,145 $42,948 Page 34 ©2024 Kaplan, Inc.  Quantitative Methods 4. Samantha Tyson must decide which of four investments are the most attractive in terms of future value. The details of each investment opportunity are as follows: 1: $1,000 annuity due with an interest rate of 7.1% and annual payments for three years. 2: $2,800 invested at an interest rate of 7.0% compounded monthly for three years. 3: $1,000 ordinary annuity with an interest rate of 7.1% and annual payments for three years. 4: $2,800 invested at an interest rate of 7.0% compounded semiannually for three years. Tyson has decided to rank each of the investments in order from highest future value to lowest. Which of the following answers correctly ranks the investments? A. 2, 4, 1, 3. B. 1, 2, 4, 3. C. 1, 2, 3, 4. D. 2, 1, 4, 3. 5. What is the value of $1,000 after 12 years at a semiannually compounded stated annual rate of 10%? A. $2,200. B. $3,138. C. $3,225. D. $3,600. 6. What is the value of $1,000 after 12 years at a quarterly compounded stated annual rate of 10%? A. $3,271. B. $3,304. C. $2,200. D. $3,385. 7. What is the value today for a lump sum of $1,000 to be received 5 years from now, using a 10% rate of interest? A. $500. B. $621. C. $667. D. $909. ©2024 Kaplan, Inc. Page 35  Quantitative Methods 8. If $5,000 is deposited into an account paying 6%, compounded monthly, what is the expected effective rate of return? A. 6.00%. B. 6.17%. C. 6.33%. D. 6.50%. 9. For any nominal rate of interest, when the number of compounding periods per year increases, the effective rate of interest: A. increases. B. decreases. C. remains the same. D. decreases at an increasing rate. 10. Cliff Bernstein is about to inherit his grandfather’s estate. Over the next twenty years Cliff will receive $17,250 at the end of each year as part of the trust set up in his name by his grandfather. In addition, Cliff will receive two one-time payments of $250,000 six years from now and $675,000 thirteen years from now. What is the present value of Cliff ’s inheritance using a 12% interest rate? A. $425,660. B. $528,117. C. $224,740. D. $410,198. 11. An investor plans to make five year-end deposits of $10,000 into an account paying 8%, compounded annually. At the end of five years (at the time of the last deposit) how much will be in the account? A. $50,000. B. $54,000. C. $58,666. D. $63,359. 12. An investor plans to make deposits of $10,000 into an account paying 8%, compounded annually. If she makes the deposits at the beginning of each year for the next five years, how much will she have in the account at the end of five years? A. $50,000. B. $54,000. C. $58,666. D. $63,359. Page 36 ©2024 Kaplan, Inc.  Quantitative Methods 13. Suppose a 30-year, $200,000 mortgage loan is taken from a bank charging 6% interest, with annual compounding. What are the 30 year-end payments? A. $6,667. B. $7,067. C. $13,707. D. $14,530. 14. An investment is expected to provide cash flows of $100 one year from today, $200 two years from today, and $500 three years from today. If the required return is 8%, the value of this investment today is closest to: A. $600. B. $661. C. $740. D. $800. 15. Mike will retire in 30 years and wants to have $2.0 million for his retirement years. Mike expects to earn 10% (compounded annually) on annual deposits to an investment account starting one year from today. How much should Mike deposit annually for each of the next 30 years to reach his goal? A. $11,010. B. $11,053. C. $12,159. D. $212,158. 16. Ben purchased a computer on credit. The store will charge an annual interest rate of 15% compounded annually. Payments are made at the end of each year. The term of the loan is two years. The annual cost to Ben is $750. How much did Ben pay for the computer? A. $1,154. B. $1,219. C. $1,304. D. $1,500. 17. Your money market account advertises a nominal rate of 2% with an effective annual rate of 2.0184%. Which of the following is the compounding frequency being used? A. Annual. B. Semiannual. C. Quarterly. D. Monthly. ©2024 Kaplan, Inc. Page 37  Quantitative Methods 18. You are considering an investment with the following expected cash flows over the next 5 years (the first cash flow is 1 year from today). Annual Cash Flow $10,000 $12,000 $14,000 X $18,000 The present value of the following cash flow stream is $44,381.54 and assumes a discount rate of 8%. Calculate the missing cash flow amount (denoted by X in the table). A. $2,000. B. $4,000. C. $8,000. D. $16,000. 19. If $5,000 is invested today and $3,000 is invested one year from today, both at the annual interest rate of 6% compounded annually, the total amount in the account two years from today is closest to: A. $8,000. B. $8,671. C. $8,798. D. $8,800. 20. Suppose your client owns a perpetuity with a present value of $1 million. What annual amount can your client withdraw every year, using an annual interest rate of 10%? A. $100,000. B. $100,100. C. $500,000. D. $1,000,000. 21. At a growth rate of 7.2%, approximately how long does it take a lump sum to double? A. 5 years. B. 1 year. C. 10 years. D. 8 years. Page 38 ©2024 Kaplan, Inc.  Quantitative Methods 22. Assume you invest a lump sum of $100 today. Further assume your investment value grows to $210 in 5 years. What is your geometric mean annual return? A. 16%. B. 20%. C. 22%. D. 25%. 23. If returns for the last five years are 2%, 15%, 17%, 19%, and 23%, which of the following represents the geometric mean? A. 12.39%. B. 14.97%. C. 15.21%. D. 18.03%. Use the following information regarding Rector, Inc. to answer Questions 24 through 26. Year Return 20x5 35% 20x6 10% 20x7 5% 20x8 –20% 20x9 35% 24. What is the arithmetic mean for the annual returns of Rector’s stock over the past five years? A. 10%. B. 11%. C. 13%. D. 15%. 25. What is the geometric mean for the annual returns of Rector’s stock over the past five years? A. 10%. B. 11%. C. 13%. D. 15%. ©2024 Kaplan, Inc. Page 39  Quantitative Methods 26. What are the range, mode, and median for the annual returns of Rector’s stock over the past five years? Range Mode Median A. 35% 10% 55% B. 55% 10% 35% C. 35% 55% 10% D. 55% 35% 10% 27. All of the following are measures of dispersion EXCEPT: A. mean average deviation. B. standard deviation. C. median. D. variance. 28. All of the following are properties of the arithmetic mean EXCEPT: A. the sum of the deviations of each value from the mean will always be zero. B. a set of data has only one mean. C. the mean is not affected by extremely large or small values. D. all values are included in computing the mean. 29. Joe has received a simple random sample of a partial year’s sales for Oryx Company. The five months of sales are as follows: $940,000, $980,000, $870,000, $940,000, and $920,000. The sample standard deviation is: A. $40,000. B. $60,000. C. $80,000. D. $100,000. 30. All of the following are correct statements about a sample EXCEPT: A. a sample should be representative of the population. B. the sample should be unbiased. C. a sample is used to estimate characteristics of the population. D. a sample includes all possible items. Page 40 ©2024 Kaplan, Inc.  Quantitative Methods Use the following information to answer Questions 31 through 36. Below are the monthly salaries for six people. Round all answers to the nearest whole number. Person Salary per Month ($) 1 5,000 2 3,000 3 3,500 4 4,000 5 4,250 6 4,300 31. Which of the following represents the arithmetic mean salary for the population? A. 4,000. B. 4,008. C. 4,200. D. 5,280. 32. Which of the following represents the mean absolute deviation of the salaries for the population? A. 508. B. 793. C. 1,492. D. 4,008. 33. Which of the following represents the variance of the salaries for the population? A. 400,347. B. 716,480. C. 741,680. D. 768,410. 34. Which of the following represents the standard deviation of the salaries for the population? A. 525.16. B. 632.73. C. 816.20. D. 861.20. ©2024 Kaplan, Inc. Page 41  Quantitative Methods 35. Which of the following represents the variance of the salaries if the data represent a sample? A. 400,347. B. 480,417. C. 890,016. D. 891,600. 36. Which of the following most closely represents the standard deviation of the salaries, if the data represent a sample? A. 632. B. 693. C. 934. D. 940. Use the following information to answer Questions 37 and 38. Assume you examine the population of oil drilling projects. The projects follow a normal distribution (e.g., a symmetric distribution) with a mean profit of $100 million and a standard deviation of $50 million. 37. What is the probability that a randomly selected oil drilling project is profitable? A. 2.5%. B. 16%. C. 95%. D. 97.5%. 38. What is the probability that a randomly selected oil drilling project will earn more than $150 million? A. 2.5%. B. 16%. C. 95%. D. 97.5%. Page 42 ©2024 Kaplan, Inc.  Quantitative Methods PRACTICE QUESTIONS ANSWERS: QUANTITATIVE METHODS m ⎛ i⎞ 1. A As illustrated in the equation ⎜⎜1+ ⎟⎟⎟ −1 , the effective interest ⎜⎝ m⎠ rate increases as the number of compounding periods per year, m, increases. As the effective rate increases, the future value increases since you are compounding at a higher rate. 2. B As the effective rate increases, the present value must decrease since you are discounting at a higher rate. 3. D When a loan is fully amortized, the payments are typically equal for the life of the loan, and each payment includes interest on the amount of the loan still outstanding (remaining principal) with the rest of the payment applied to the principal balance. The payment is found using the ordinary annuity method as follows: –100,000 PV 5 I/Y 5 N CPT PMT = $23,097 Interest in each year equal the interest rate (5.0%) times the principal balance at the end of the previous year. For the third year, the interest portion of the payment is 0.05 × 62,900 = $3,145. The principal portion of the payment is 23,097 – 3,145 = $19,952. Thus the principal balance gets reduced to 62,900 – 19,952 = $42,948 at the end of year three. The following amortization table demonstrates the interest, principal, and outstanding balance for each of the five years the loan is outstanding. Year Payment Interest Principal Balance 0 100,000 1 23,097 5,000 18,097 81,903 2 23,097 4,095 19,002 62,900 3 23,097 3,145 19,952 42,948 4 23,097 2,147 20,950 21,998 5 23,097 1,100 21,998 0 ©2024 Kaplan, Inc. Page 43  Quantitative Methods 4. D Begin by calculating the future value of each investment as follows: Investment 1: Calculator keystrokes: 2nd ⟹ BGN 2nd ⟹ SET 2nd ⟹ QUIT –1,000 PMT 7.1 I/Y 3 N CPT FV = $3,447 Investment 2: Calculator keystrokes: –2,800 PV 7 ÷ 12 I/Y 3 × 12 N CPT FV = $3,452 Investment 3: Calculator keystrokes: –1,000 PMT 7.1 I/Y 3 N CPT FV = $3,218 Investment 4: Calculator keystrokes: –2,800 PV 7÷2 I/Y 3×2 N CPT FV = $3,442 Therefore the investments in order of largest to smallest future value are 2, 1, 4, 3. For Questions 5 and 6 use the following formula: m×n ⎛ i ⎟⎞ ⎜ FVn = PV ⎜1 + ⎟⎟ ⎜⎝ m⎠ where: m = number of compounding periods per year i = stated or nominal rate of interest n = number of years PV = beginning amount or present value FV = ending amount or future value Page 44 ©2024 Kaplan, Inc.  Quantitative Methods Alternatively, you can plug the values into your TI BA II Plus as shown. If you prefer to do that, set your P/Y value to 1.0 and enter the values. Remember, the calculator will assume one cash flow is an inflow and the other an outflow. 5. C FV = 1,000(1 + 0.10/2)12×2 = 3,225.10, or PV = 1,000, years = 12, compounding periods per year = 2, interest rate = 10% Calculator keystrokes: –1,000 PV 10 ÷ 2 I/Y 12 × 2 N CPT FV = $3,225.10 6. A FV = 1000(1 + 0.10/4)12×4 = $3,271.49, or PV = 1000, years = 12, compounding periods per year = 4, interest rate = 10% Calculator keystrokes: –1,000 PV 10 ÷ 4 I/Y 12 × 4 N CPT FV = $3,271.49 1000 7. B PV = = 620.92, or (1.10)5 FV = 1000, years = 5, compounding periods per year = 1, interest rate = 10% Calculator keystrokes: 1,000 FV 10 I/Y 5 N CPT PV = $620.92 8. B The effective return for an account is the same regardless of the amount of money deposited. Using the effective interest rate formula: ⎛ ⎞m ⎜⎜1+ i ⎟⎟ −1.0 = 12, i = 0.06, effective rate = 1.0617 – 1 = 6.17% ⎜⎝ m ⎟⎠ 9. A For any nominal rate of interest, when the number of compounding periods per year increases, the effective rate of interest will increase. ©2024 Kaplan, Inc. Page 45  Quantitative Methods 10. D The key to a problem such as this one is to find the present value of each investment separately and then add the present values together. There are three investments in this problem: an annuity of $17,250 that will last 20 years, a lump sum payment of $250,000 that will occur in six years, and a lump sum payment of $675,000 that will occur in thirteen years. We find the present values as follows: Annuity (make sure the calculator is in END mode): Calculator keystrokes: –17,250 PMT 12 I/Y 20 N CPT PV = $128,848 First Lump Sum: –250,000 FV 12 I/Y 6 N CPT PV = $126,658 Second Lump Sum: –675,000 FV 12 I/Y 13 N CPT PV = $154,693 Thus, the total value of the inheritance is $128,848 + $126,658 + $154,693 = $410,198. 11. C These are end-of-year payments. Your calculator should be set to END mode. Calculator keystrokes: –10,000 PMT 8 I/Y 5 N CPT FV = $58,666.01 12. D These are beginning-of-year payments. Your calculator should be set to BGN mode. Calculator keystrokes: –10,000 PMT 5 N 8 I/Y CPT FV = $63,359.29 Page 46 ©2024 Kaplan, Inc.  Quantitative Methods 13. D The present value of the payments must equal the amount borrowed. Be sure your calculator is set to END mode. Calculator keystrokes: –200,000 PV 6 I/Y 30 N CPT PMT = $14,529.78 100 200 500 14. B PV = + + = 92.59 +171.47 + 396.92 = $660 6.98 (1.08)1 (1.08)2 (1.08)3 15. C Be sure your calculator is set to END mode. Calculator keystrokes: 2,000,000 FV 10 I/Y 30 N CPT PMT = $12,158.50 16. B Be sure your calculator is set to END mode. Calculator keystrokes: 750 PMT 15 I/Y 2 N CPT PV = $1,219.28 m m ⎛ i⎞ ⎛ 0.02 ⎟⎞ 17. D Effective return = 0.02 = ⎜⎜⎜1+ ⎟⎟⎟ − 1 = ⎜⎜⎜1+ ⎟⎟ −1 ⎝ m⎠ ⎝ m ⎠ To use trial and error, plug different choices for m: annual = 1, gives an effective rate of 2%, easy choice to eliminate semiannual = 2, gives an effective rate of 2.01%, not the correct answer quarterly = 4, gives an effective rate of 2.0151%, not the correct answer monthly = 12; gives an effective rate of 2.0184%, the correct answer